10 Bessel Functions Modified Bessel Functions 10.36 Other Differential Equations 10.38 Derivatives with Respect to Order

§10.37 Inequalities; Monotonicity

ⓘ

Keywords:: inequalities, modified Bessel functions, monotonicity
Notes:: See Olver (1997b, pp. 251–252). For (10.37.1) see Everitt and Jones (1977, (8.5)) which contains an error. Formula (8.5) holds for $|\theta|\leq\tfrac{1}{2}\pi$ only. Hence the sector of validity of (10.37.1) is not $|\operatorname{ph}z|<\pi$ as was previously mentioned.
Referenced by:: Erratum (V1.0.11) for References, Erratum (V1.0.17) for Section 10.37, Erratum (V1.0.17) for Section 10.37, Erratum (V1.0.7) for References
Permalink:: http://dlmf.nist.gov/10.37
Errata (effective with 1.0.17):: For (10.37.1), it was originally stated incorrectly that (10.37.1) holds for $|\operatorname{ph}z|<\pi$ . See Everitt and Jones (1977, (8.5)), which makes this incorrect claim.
Reported 2017-11-14 by Gergő Nemes
Addition (effective with 1.0.11):: The reference to Gaunt (2014) was added.
Addition (effective with 1.0.7):: The reference to Segura (2011) has been added at the end of this section.
See also:: Annotations for Ch.10

If $\nu$ $(\geq 0)$ is fixed, then throughout the interval $0<x<\infty$ , $I_{\nu}\left(x\right)$ is positive and increasing, and $K_{\nu}\left(x\right)$ is positive and decreasing.

If $x$ $(>0)$ is fixed, then throughout the interval $0<\nu<\infty$ , $I_{\nu}\left(x\right)$ is decreasing, and $K_{\nu}\left(x\right)$ is increasing.

For sharper inequalities when the variables are real see Paris (1984) and Laforgia (1991).

If $0\leq\nu<\mu$ and $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi$ , then

10.37.1		$\|K_{\nu}\left(z\right)\|<\|K_{\mu}\left(z\right)\|.$
ⓘ Symbols: $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $z$ : complex variable and $\nu$ : complex parameter Referenced by: §10.37, §10.37, Erratum (V1.0.17) for Section 10.37 Permalink: http://dlmf.nist.gov/10.37.E1 Encodings: TeX, pMML, png See also: Annotations for §10.37 and Ch.10

Note that previously we did mention that (10.37.1) holds for $|\operatorname{ph}z|<\pi$ . This is definitely not the case.

See also Pal^′tsev (1999), Petropoulou (2000), Segura (2011) and Gaunt (2014).

10.36 Other Differential Equations 10.38 Derivatives with Respect to Order

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